Geoscience Reference
In-Depth Information
Q
A
,
M
+
Q
B
,
N
−
Q
C
,1
δ
Q
C
,1
−
δ
Q
A
,
M
−
δ
Q
B
,
N
=
(5.85)
Substituting Eqs. (5.79) and (5.80) into Eq. (5.85) and then using the expressions
for
δ
h
A
,
M
and
δ
h
B
,
N
obtained from Eqs. (5.83) and (5.84) yields
δ
Q
C
,1
=
S
C
,1
δ
h
C
,1
+
T
C
,1
(5.86)
Q
A
,
M
+
Q
B
,
N
−
Q
C
,1
+
z
sC
,1
−
z
sA
,
M
)
+
where
S
C
,1
=
S
A
,
M
+
S
B
,
N
, and
T
C
,1
=
S
A
,
M
(
z
sC
,1
−
z
sB
,
N
)
+
S
B
,
N
T
B
,
N
.
The forward sweep can then be carried out from the first to the last point in channel
C
using Eqs. (5.71) and (5.72).
The return sweep starts from the last point of channel
C
, at which a boundary con-
dition is specified. The stage and discharge increments at the last point are determined
using Eqs. (5.75)-(5.78), and at the intermediate points using Eqs. (5.69) and (5.74).
At the end of the return sweep in channel
C
back to the junction, the stage increment
δ
(
T
A
,
M
+
h
C
,1
is calculated using Eq. (5.74), and the discharge increment
δ
Q
C
,1
is determined
using Eq. (5.86). Next, the stage increments
δ
h
A
,
M
and
δ
h
B
,
N
are determined using
Eqs. (5.83) and (5.84), and the discharge increments
Q
B
,
N
are computed
using Eqs. (5.79) and (5.80). Finally, the return sweep can be carried out along both
channels
A
and
B
.
It should be noted that the equal water stage condition (5.81) may be replaced with
the equal energy level condition at the junction:
δ
Q
A
,
M
and
δ
Q
n
+
1
A
,
M
A
n
+
1
A
,
M
Q
n
+
1
B
,
N
A
n
+
1
B
,
N
Q
n
+
1
C
,1
A
n
+
1
C
,1
2
2
2
1
2
g
1
2
g
1
2
g
z
n
+
1
sA
,
M
z
n
+
1
sB
,
N
z
n
+
1
sC
,1
+
=
+
=
+
(5.87)
which can be expanded in terms of
δ
h
and
δ
Q
and used to substitute Eqs. (5.83)
and (5.84).
Algorithm for a looped channel network
A “looped” channel network is shown in Fig. 5.12. The difference between “den-
dritic” and “looped” channel networks is that there is only one possible flow path
from a given point to another in a dentritic network, while there are usually sev-
eral such flow paths in a looped network. The previous double sweep algorithm
cannot be applied directly to the solution of unsteady flows in looped channel net-
works. The looped solution algorithm described by Cunge
et al
. (1980) is often used
instead. In this algorithm, the term “node” is used to represent the junction of several
flow paths that originate from either other nodes or boundary points. For exam-
ple, the nodes in the channel network shown in Fig. 5.12 are
A
,
B
,
C
, and
D
. The
points (cross-sections) between two nodes in each channel are defined as intermediate
points.
